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Mathematics Awareness Month Theme Essay

The Shape of the Universe

Early Models of the Shape of the Universe

For thousands of years, people believed that the universe revolved around the earth and astronomers created mathematical models in order to explain observations in the sky. Eudoxus created a model containing rotating spheres centered about the earth. Aristotle adopted and described this model. While he was able to partially explain some of the planetary motions by rotating the spheres at different velocities, other observations, such as differences in brightness levels, could not be resolved. In his famous work, the Almagest, Claudius Ptolemy, a 2nd century philosopher, refined and improved the model based on the earlier work of Apollonius and Hipparchus. In the Ptolemaic Universe planets now moved along epicycles (see the related JavaSketch for more information [8]), which had circles attached to the spheres around the earth, and yet this model still did not completely resolve earlier difficulties.

Aristarchus had suggested a heliocentric system and in the 16th century, Nicolaus Copernicus gave substance to Aristarchus' ideas by carrying out the detailed mathematical calculations:

Finally we shall place the Sun himself at the center of the Universe. All this is suggested by the systematic procession of events and the harmony of the whole Universe, if only we face the facts, as they say, 'with both eyes open'.

Copernicus, De Revolutionibus Orbium Coelestium, 1543

His model still utilized epicycles in order to explain the circular motion of the planets, but it placed a motionless sun close to the center of the universe.

Johannes Kepler revolutionized astronomy by finally overthrowing the stranglehold of purely circular motions. His introduction of elliptical orbits, together with his other two laws of planetary motion, form the basis of celestial mechanics to this day. They were also critical in the formulation and verification of Sir Isaac Newton's laws of gravity and of motion, which in turn became the basis for cosmology for the following two centuries.

In 1917, Albert Einstein used Bernhard Riemann's mathematics in order to present a model for the universe that was consistent with his theory of relativity. His model was based on a finite spherical universe that obeyed different laws than those of a flat Euclidean universe [6]. Since then mathematicians and astronomers have debated whether the universe is Euclidean or non-Euclidean, and whether it is finite or infinite.

Euclidean or Non-Euclidean?

Carl Friedrich Gauss performed an experiment to measure the angles of a triangle formed by light rays whose vertices were between Brocken, Hohenhagen, and Inselberg, three mountain peaks in Germany. Since Euclidean triangles have 180 degrees as the measure of the sum of the angles (see the related JavaSketch for more information), it might seem that this method could determine whether the universe is Euclidean. Even if the universe is non-Euclidean, a triangle of this size would have been too small to detect the possible curvature within the accuracy of the measuring instruments, and light rays are not good measures of shortest distance paths because they bend with gravity. As far as we know, while it is true that Gauss did perform measurements on this triangle, it is a myth that he wanted to use these measurements to determine whether the universe is Euclidean [1]. Instead, it was Nikolai Lobachevsky who proposed to measure the triangle formed by the star Sirius and the position of the earth at different times, 6 months apart, in order to determine the curvature of space [7]. The Sirius measurements were inconclusive because they were close to 180 degrees, within the expected margin of error.

In 1900, Karl Schwarzschild conducted a detailed analysis of the amount the universe would have to be positively or negatively curved in order for the known parallax measurements at the time to be Euclidean within the margin of error then available. Today, astronomers still wonder whether our universe is Euclidean or not. In the foreseeable future, we can only measure triangles with an area less than that of our solar system, but we would need to look further than that to determine the geometry [6].

Finite or Infinite?

There is also disagreement over whether the universe is finite or infinite. Astronomers have noticed a cosmic microwave background radiation coming from all directions of space. There are slight variations that may enable us to determine the shape of space, although none of the observations so far have been accurate enough to make a definite determination. Recent probes such as NASA's WMAP (Wilkinson Microwave Anisotropy Probe) and a planned 2007 launch of the Planck satellite hope to do so.

Astronomers have analyzed the WMAP data and they have obtained conflicting results. Jean-Pierre Luminet and his colleagues proposed that the data seemed to best fit a universe that was a spherical space formed by identifying opposite faces of a dodecahedron in a three-dimensional sphere [10]. You can build a dodecahedron, a polyhedron with 12 pentagonal faces, to see that the faces cannot be glued straight across without first using a twist. Other mathematicians and physicists, such as Max Tegmark and his colleagues, assert that the WMAP data in fact rules out a finite universe, and that measurements point to a flat Euclidean space which is infinite [11].

Understanding the Finite Theory

Our universe could be finite but still have no edges. The first to make that observation was Riemann, with his proposal of a spherical universe. As another example, a spaceship traveling off one side of the screen below will reappear on the opposite side, creating the illusion of traveling in an infinite space. To see what finite shape this actually is, you can start with a square piece of stretchy material and glue together the left and right sides to obtain a cylinder. Next, identify the top and bottom sides and the cylinder will fold up into a torus. In a similar way, identify opposite faces of a cube or a polyhedron in higher dimensions to obtain a possible shape for our universe [14].

 A Finite 2-D Torus Universe [12]

You can experience what it is like to live on a 2-D torus universe by playing Torus Tic-Tac-Toe [12]. In the game, the top left square is really next to the top right square. You are allowed to "scroll" the board in order to help develop your intuition (once a square has been labeled X or O, you can click on it, hold it down, and move the board around to see the identifications).

To visualize a finite universe in three dimensions, you can watch the Futurama [4] episode I, Roommate (Season 1 DVD):

Fry and Bender are looking for housing. Leela, Fry, Bender and the manager enter an apartment that resembles Dutch graphic artist M.C. Escher's Relativity print [3].
Fry: I'm not sure we wanna pay for a dimension we're not gonna use.
Bender, the robot, falls down the staircase and continues to fall "down" the other staircases in many different directions.

Look for Bender in each of the frames, and use his position to give gluing instructions and explain which openings are identified [5].

Regardless of whether our universe is finite or infinite, the classification of finite universes is an area of intense interest to mathematicians. There are ten closed three-dimensional Euclidean universes, glued together by identifying the opposite faces of a cube, just like the above Futurama apartment. The number of spherical possibilities are infinite, but they have been classified completely, and one of them is the dodecahedral universe mentioned by Luminet. There are infinitely many possibilities for a hyperbolic universe, another non-Euclidean geometry [2], and their rich structure and classification is still the subject of research today.

Conclusion

The quest to understand the precise geometry and shape of our universe began thousands of years ago, when mathematicians and astronomers used mathematical models to try and explain their observations. While there seem to be some irregularities in the WMAP data that throw the conflicting analyses and conclusions into doubt [13], there is hope that the data from the proposed 2007 Planck satellite will ultimately lead us to the answer. Even if it does not, we will continue to develop new models and methods so that one day we can determine the shape of space.

The author gratefully acknowledges the input of Robert Osserman of MSRI.

References

1. Breitenberger, Ernst, Gauss's Geodesy and the Axiom of Parallels, Archive for the History of the Exact Sciences, 31 (1984), pp. 273-289.
2. Dunham, Douglas, Hyperbolic Art and the Poster Pattern, http://mathaware.org/mam/03/essay1.html.
3. Escher, M.C., Relativity, 1953.
4. Futurama TM and copyright Twentieth Century Fox and its related companies, http://www2.foxstore.com/detail.html?item=606.
5. Greenwald, Sarah J., Classroom Activities on the Geometry of the Earth and Universe, http://www.mathsci.appstate.edu/~sjg/talks/earthanduniverse.html.
6. Henderson, David and Daina Taimina, Experiencing Geometry: Euclidean and Non-Euclidean with Strands of History, Prentice Hall, 2004.
7. Jammer, Max, Concepts of Space: A History of Theories of Space in Physics, Dover, 1969, p. 150.